3.1030 \(\int (a+b x) (a c-b c x) \, dx\)

Optimal. Leaf size=18 \[ a^2 c x-\frac{1}{3} b^2 c x^3 \]

[Out]

a^2*c*x - (b^2*c*x^3)/3

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Rubi [A]  time = 0.0042267, antiderivative size = 18, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067, Rules used = {41} \[ a^2 c x-\frac{1}{3} b^2 c x^3 \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)*(a*c - b*c*x),x]

[Out]

a^2*c*x - (b^2*c*x^3)/3

Rule 41

Int[((a_) + (b_.)*(x_))^(m_.)*((c_) + (d_.)*(x_))^(m_.), x_Symbol] :> Int[(a*c + b*d*x^2)^m, x] /; FreeQ[{a, b
, c, d, m}, x] && EqQ[b*c + a*d, 0] && (IntegerQ[m] || (GtQ[a, 0] && GtQ[c, 0]))

Rubi steps

\begin{align*} \int (a+b x) (a c-b c x) \, dx &=\int \left (a^2 c-b^2 c x^2\right ) \, dx\\ &=a^2 c x-\frac{1}{3} b^2 c x^3\\ \end{align*}

Mathematica [A]  time = 0.0016718, size = 18, normalized size = 1. \[ c \left (a^2 x-\frac{b^2 x^3}{3}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)*(a*c - b*c*x),x]

[Out]

c*(a^2*x - (b^2*x^3)/3)

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Maple [A]  time = 0., size = 17, normalized size = 0.9 \begin{align*}{a}^{2}cx-{\frac{{b}^{2}c{x}^{3}}{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)*(-b*c*x+a*c),x)

[Out]

a^2*c*x-1/3*b^2*c*x^3

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Maxima [A]  time = 1.02333, size = 22, normalized size = 1.22 \begin{align*} -\frac{1}{3} \, b^{2} c x^{3} + a^{2} c x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(-b*c*x+a*c),x, algorithm="maxima")

[Out]

-1/3*b^2*c*x^3 + a^2*c*x

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Fricas [A]  time = 1.24425, size = 35, normalized size = 1.94 \begin{align*} -\frac{1}{3} x^{3} c b^{2} + x c a^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(-b*c*x+a*c),x, algorithm="fricas")

[Out]

-1/3*x^3*c*b^2 + x*c*a^2

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Sympy [A]  time = 0.057572, size = 15, normalized size = 0.83 \begin{align*} a^{2} c x - \frac{b^{2} c x^{3}}{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(-b*c*x+a*c),x)

[Out]

a**2*c*x - b**2*c*x**3/3

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Giac [A]  time = 1.05493, size = 22, normalized size = 1.22 \begin{align*} -\frac{1}{3} \, b^{2} c x^{3} + a^{2} c x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(-b*c*x+a*c),x, algorithm="giac")

[Out]

-1/3*b^2*c*x^3 + a^2*c*x